Comparison of two ontogenetic growth equations for animals and plants

Comparison of four performance models in quantifying the inequality of leaf and fruit size distribution

The inequality in leaf and fruit size distribution per plant can be quantified using the Gini index, which is linked to the Lorenz curve depicting the cumulative proportion of leaf (or fruit) size against the cumulative proportion of the number of leaves (or fruits). Prior researches have predominantly employed empirical models-specifically the original performance equation (PE-1) and its generalized counterpart (GPE-1)-to fit rotated and right-shifted Lorenz curves. Notably, another potential performance equation (PE-2), capable of generating similar curves to PE-1, has been overlooked and not systematically compared with PE-1 and GPE-1. Furthermore, PE-2 has been extended into a generalized version (GPE-2). In the present study, we conducted a comparative analysis of these four performance equations, evaluating their applicability in describing Lorenz curves related to plant organ (leaf and fruit) size. Leaf area was measured on 240 culms of dwarf bamboo (Shibataea chinensis Nakai), and fruit volume was measured on 31 field muskmelon plants (Cucumis melo L. var. agrestis Naud.). Across both datasets, the root-mean-square errors of all four performance models were consistently smaller than 0.05. Paired t-tests indicated that GPE-1 exhibited the lowest root-mean-square error and Akaike information criterion value among the four performance equations. However, PE-2 gave the best close-to-linear behavior based on relative curvature measures. This study presents a valuable tool for assessing the inequality of plant organ size distribution.

'biogeom': An R package for simulating and fitting natural shapes

Many natural objects exhibit radial or axial symmetry in a single plane. However, a universal tool for simulating and fitting the shapes of such objects is lacking. Herein, we present an R package called 'biogeom' that simulates and fits many shapes found in nature. The package incorporates novel universal parametric equations that generate the profiles of bird eggs, flowers, linear and lanceolate leaves, seeds, starfish, and tree-rings, and three growth-rate equations that generate the profiles of ovate leaves and the ontogenetic growth curves of animals and plants. 'biogeom' includes several empirical datasets comprising the boundary coordinates of bird eggs, fruits, lanceolate and ovate leaves, tree rings, seeds, and sea stars. The package can also be applied to other kinds of natural shapes similar to those in the datasets. In addition, the package includes sigmoid curves derived from the three growth-rate equations, which can be used to model animal and plant growth trajectories and predict the times associated with maximum growth rate. 'biogeom' can quantify the intra- or interspecific similarity of natural outlines, and it provides quantitative information of shape and ontogenetic modification of shape with important ecological and evolutionary implications for the growth and form of the living world.

The Modified Brière Equation and Its Applications

The Brière equation (BE) is widely used to describe the effect of temperature on the development rate of insects, and it can produce both symmetrical and asymmetrical bell-shaped curves. Because of its elasticity in curve fitting, the integrated form of BE has been recommended for use as a sigmoid growth equation to describe the increase in plant biomass with time. However, the start time of growth predicted by the sigmoid growth equation based on the BE is not completely comparable to empirical crop growth data. In the present study, we modified the BE by adding an additional parameter to further increase its elasticity for data fitting. We termed this new equation the modified Brière equation (MBE). Data for the actual height and biomass of 15 species of plants (with two cultivars for one species) were fit with the sigmoid growth equations based on both the BE and MBE assuming that the growth start time was zero for both. The goodness of fit of the BE and MBE sigmoid growth equations were compared based on their root-mean-square errors and the corresponding absolute percentage error between them when fit to these data. For most species, we found that the MBE sigmoid growth equation achieved a better goodness of fit than the BE sigmoid growth equation. This work provides a useful tool for quantifying the ontogenetic or population growth of plants.

Plant functional traits shape growth rate for xerophytic shrubs

Trade-offs exist for xerophytic shrubs between functional traits, involving in water loss and assimilate accumulation, can contribute to its survival and growth rate regulation in arid environments. However, growth analysis based on plant functional traits has been focused on the study of herbs and woody species. It is still unclear how the functional traits of xerophytic shrubs regulate their growth rate. In this study, we selectedeight xerophytic shrubs as samples to analyze the regulation process of the functional traits of shrubs on growth rate. Plants were cultivated for three years, and three harvests (every one year) were carried out. Factors explaining between-species differences in relative growth rate (RGR) varied, depending on whether different ages were considered. The results showed that RGR was positively correlated with net assimilation rate, but there was a significant negative correlation with leaf area ration (LAR), specific leaf area (SLA), and leaf biomass ratio in the age 1. However, in the age 2, RGR showed a significant positive correlation with the morphological traits (i.e., leaf area ration and specific leaf area), but not with physiological traits (i.e., net assimilation rate) and leaf biomass allocation. Our results suggested that the fluctuation of environmental factors affects the regulation path of the plant functional traits on RGR of xerophytic shrubs. However, the analysis of causality model showed that no matter in which age, net assimilation rate and leaf area ration principally drive the variation in RGR among xerophytic shrubs.

Effects of environmental parameters on Lemna minor growth: An integrated experimental and modelling approach

Pollution of surface waters is a worldwide problem for people and wildlife. Remediation and phytoremediation approaches can offer a solution to deal with specific scenarios. Lemna minor, commonly known as duckweed, can absorb and accumulate pollutants in its biomass. To evaluate if L. minor could be applied for phytoremediation purposes, it is necessary to further investigate its remediation capability and to identify which parameters affect the remediation process. Such a model must include both plant growth and pollutant exchange. A remediation model based on a robust experimental study can help to evaluate L. minor as a proper remediation strategy and to predict the outcome of a L. minor based remediation system. To set up this model, this paper focusses on a detailed experimental study and a comprehensive mathematical modelling approach to represent L. minor growth as a function of biomass, temperature, light irradiation and variable nutrient concentrations. The influence of environmental conditions on L. minor growth was studied, by composing 7 days growth curves. Plants were grown under predefined environmental conditions (25°C, 14h photoperiod, 220 μmol m^-2 s^-1 light intensity and a modified Hoagland solution with 23.94 mg N L^-1 and 3.10 mg P L^-1 (N:P ratio of 7.73)) as standard for all experiments. The influence of different temperatures (6, 10, 15, 20, 25, 30 and 35°C), light intensities (63, 118, 170, 220 and 262 μmol m^-2 s^-1), photoperiods (12h and 14h) and N:P ratios (1.18, 3.36, 7.73 and 29.57) were tested in the model. As a result, a growth model was optimised using separate datasets for temperature, light intensity, photoperiod and nutrients and validated by further integrated testing. The growth model is a stable platform for application in phytoremediation of radionuclides in contaminated water, to be extended in future studies with information of pollutant uptake, pollutant-nutrient interactions and transfer to the biomass.

A General Model for Describing the Ovate Leaf Shape

Many plant species produce ovate leaves, but there is no general parametric model for describing this shape. Here, we used two empirical nonlinear equations, the beta and Lobry–Rosso–Flandrois (LRF) equations, and their modified forms (referred to as the Mbeta and MLRF equations for convenience), to generate bilaterally symmetrical curves along the x-axis to form ovate leaf shapes. In order to evaluate which of these four equations best describes the ovate leaf shape, we used 14 leaves from 7 Neocinnamomum species (Lauraceae) and 72 leaves from Chimonanthus praecox (Calycanthaceae). Using the AIC and adjusted root mean square error to compare the fitted results, the modified equations fitted the leaf shapes better than the unmodified equations. However, the MLRF equation provided the best overall fit. As the parameters of the MLRF equation represent leaf length, maximum leaf width, and the distance from leaf apex to the point associated with the maximum leaf width along the leaf length axis, these findings are potentially valuable for studying the influence of environmental factors on leaf shape, differences in leaf shape among closely related plant species with ovate leaf shapes, and the extent to which leaves are bilaterally symmetrical. This is the first work in which temperature-dependent developmental equations to describe the ovate leaf shape have been employed, as previous studies lacked similar leaf shape models. In addition, prior work seldom attempted to describe real ovate leaf shapes. Our work bridges the gap between theoretical leaf shape models and empirical leaf shape indices that cannot predict leaf shape profiles.

Spatial and temporal autocorrelations affect Taylor's law for US county populations: Descriptive and predictive models

Understanding the spatial and temporal distributions and fluctuations of living populations is a central goal in ecology and demography. A scaling pattern called Taylor's law has been used to quantify the distributions of populations. Taylor's law asserts a linear relationship between the logarithm of the mean and the logarithm of the variance of population size. Here, extending previous work, we use generalized least-squares models to describe three types of Taylor's law. These models incorporate the temporal and spatial autocorrelations in the mean-variance data. Moreover, we analyze three purely statistical models to predict the form and slope of Taylor's law. We apply these descriptive and predictive models of Taylor's law to the county population counts of the United States decennial censuses (1790-2010). We find that the temporal and spatial autocorrelations strongly affect estimates of the slope of Taylor's law, and generalized least-squares models that take account of these autocorrelations are often superior to ordinary least-squares models. Temporal and spatial autocorrelations combine with demographic factors (e.g., population growth and historical events) to influence Taylor's law for human population data. Our results show that the assumptions of a descriptive model must be carefully evaluated when it is used to estimate and interpret the slope of Taylor's law.

Application of an Ovate Leaf Shape Model to Evaluate Leaf Bilateral Asymmetry and Calculate Lamina Centroid Location

Leaf shape is an important leaf trait, with ovate leaves common in many floras. Recently, a new leaf shape model (referred to as the MLRF equation) derived from temperature-dependent bacterial growth was proposed and demonstrated to be valid in describing leaf boundaries of many species with ovate leaf shape. The MLRF model's parameters can provide valuable information of leaf shape, including the ratio of lamina width to length and the lamina centroid location on the lamina length axis. However, the model wasn't tested on a large sample of a single species, thereby limiting its overall evaluation for describing leaf boundaries, for evaluating lamina bilateral asymmetry and for calculating lamina centroid location. In this study, we further test the model using data from two Lauraceae species, Cinnamomum camphora and Machilus leptophylla, with >290 leaves for each species. The equation was found to be credible for describing those shapes, with all adjusted root-mean-square errors (RMSE) smaller than 0.05, indicating that the mean absolute deviation is smaller than 5% of the radius of an assumed circle whose area equals lamina area. It was also found that the larger the extent of lamina asymmetry, the larger the adjusted RMSE, with approximately 50% of unexplained variation by the model accounted for by the lamina asymmetry, implying that this model can help to quantify the leaf bilateral asymmetry in future studies. In addition, there was a significant difference between the two species in their centroid ratio, i.e., the distance from leaf petiole to the point on the lamina length axis associated with leaf maximum width to the leaf maximum length. It was found that a higher centroid ratio does not necessarily lead to a greater investment of mass to leaf petiole relative to lamina, which might depend on the petiole pattern.

Internode morphometrics and allometry of Tonkin Cane Pseudosasa amabilis

Pseudosasa amabilis (McClure) (Poales: Gramineae) is a typical bamboo species naturally distributed in large area of south China and famous for its culm strength. Although bamboos were found to share the same development rule, the detailed internode morphology of bamboo culm was actually not fully expressed. We explored internode morphology of P. amabilis using 11 different physical parameters in different dimensions (1–4). As Taylor's power law (TPL) is generally applicable to describe relationship between mean and variance of population density, here we used TPL to evaluate the differences between internodes, and further, the relationship between dimension and TPL. Results showed that length (L), hollow radius (HR), hollow area (HA), hollow cylinder volume (HCV), total cylinder volume (TCV), density (De), and weight (W) all presented positive skewed distribution in varying degrees. For the basic one‐dimensional parameters, the 9th internode was the longest, the 7th the heaviest, while thickness (T) decreased with internodes. Diameter (D) decreased in general but with an inconspicuous local mode at the 5–6th internodes, potentially due to the rapid height growth. The longest (9th) internode was the “turning point” for T‐D and HR‐D relationships. Scatter plot changing trends of W to the one‐dimensional parameters after the heaviest (7th) internode were reversed, indicating a deceleration of growth speed. TPL was not holding well in one‐dimensional parameters (R²: 0.5413–0.8125), but keep increasing as the parameter's dimension increasing (R² > 0.92 for two‐dimensional, R² > 0.97 for three‐dimensional, and R² > 0.99 for four‐dimensional parameters.), suggesting an emergence mechanism of TPL related to both the physical dimensions of morphological measures and the allometric growth of bamboo. From the physical fundamental level, all existences are the expression of energy distribution in different dimensions, implying a more general rule that energy distribution holds better TPL in higher dimension level.